Báo cáo toán học: "The Edmonds-Gallai Decomposition for the k-Piece Packing Problem" potx

The Edmonds-Gallai Decomposition for the k-Piece
Packing Problem
Marek Janata
Dept. of Applied Mathematics
and Institute of Theoretical Computer Science (ITI),
Charles University, Malostranske n. 25,
118 00 Praha 1, Czech Republic.
,
Martin Loebl
Dept. of Applied Mathematics
and Institute of Theoretical Computer Science (ITI),
Charles University, Malostranske n. 25,
118 00 Praha 1, Czech Republic.
,and
J´acint Szab´o
∗
Dept. of Operations Research, E¨otv¨os University,
P´azm´any P´eter s´et´any 1/C,
Budapest, Hungary H-1117.

Submitted: Feb 4, 2004; Accepted: Feb 4, 2005; Published: Feb 14, 2005
MR Subject Classifications: 05C70
Abstract
Generalizing Kaneko’s long path packing problem, Hartvigsen, Hell and Szab´o
consider a new type of undirected graph packing problem, called the k-piece pack-
ing problem.Ak-piece is a simple, connected graph with highest degree exactly k
so in the case k = 1 we get the classical matching problem. They give a polyno-
mial algorithm, a Tutte-type characterization and a Berge-type minimax formula
for the k-piece packing problem. However, they leave open the question of an
Edmonds-Gallai type decomposition. This paper fills this gap by describing such
a decomposition. We also prove that the vertex sets coverable by k-piece packings

have a certain matroidal structure.
∗
Research is supported by OTKA grants T 037547, N 034040 and by the Egerv´ary Research Group
of the Hungarian Academy of Sciences and by European MCRTN Adonet, Contract Grant No. 504438.
the electronic journal of combinatorics 12 (2005), #R8 1
1 Introduction
In this paper all graphs are simple and undirected. Given a set F of graphs, an F-packing
ofagraphG is a subgraph P of G such that each connected component of P is isomorphic
to a member of F.AnF-packing P is called maximal if there is no F-packing P

with
V (P )  V (P

). An F-packing is maximum if it covers a maximum number of vertices of
G and it is perfect if it covers every vertex of G.TheF-packing problem is to describe
the properties of the F-packings of G. Finally, the F-packing problem is polynomial if
for all input graphs G the size of the maximum F-packings of G can be determined in
time polynomial in the size of G. (The size of a graph is the number of its vertices.)
Several polynomial F-packing problems are known in the case K
2
∈F. For instance,
we get a polynomial packing problem if F consists of K
2
and a finite set of hypomatchable
graphs [2, 3, 4, 6]. A complete classification of the {K
2
,F}-packing problems for graphs
F is given in [10]. In all known polynomial F-packing problems with K
2
∈Fit holds

that each maximal F-packing is maximum too; those vertex sets which can be covered
by an F-packing form a matroid (this is the matroidal property); and the analogue of the
Edmonds-Gallai structure theorem holds.
The first polynomial F-packing problem with K
2
/∈F was considered by Kaneko [7],
who presented a Tutte-type characterization of graphs having a perfect packing by long
paths, ie. by paths of length at least 2. A shorter proof for Kaneko’s theorem and a min-
max formula was subsequently found by Kano, Katona and Kir´aly [8] but polynomiality
remained open. The long path packing problem was generalized by Hartvigsen, Hell and
Szab´o [5] by introducing the k-piece packing problem,ie.theF-packing problem where
F consists of all connected graphs with highest degree exactly k. Such a graph is called a
k-piece. Note that a 1-piece is just K
2
, thus the 1-piece packing problem is the classical
matching problem. The 2-piece packing problem is equivalent to the long path packing
problem because a 2-piece is either a long path or a circuit C of length at least 3 so deleting
an edge from C results in a long path. The main result of [5] is a polynomial algorithm for
finding a maximum k-piece packing. From this algorithm a characterization for graphs
having a perfect k-piece packing and a min-max result for the size of a maximum k-piece
packing are derived.
Neither the Edmonds-Gallai decomposition nor the matroidal property of packings is
considered in [5]. This paper fills this gap by giving a canonical Edmonds-Gallai type de-
composition for the k-piece packing problem. We also show that the vertex sets coverable
by maximal k-piece packings have a certain matroidal structure, see Section 2. It turns
outthatinthek-piece packing problem maximal and maximum packings do not coincide
and the maximal packings are of more interest than the maximum ones.
In Section 5 we present some results on barriers related to k-piece packings, for instance
we prove that the intersection of two barriers is a barrier.
The number of connected components of a graph G is denoted by c(G) and the highest

degree of G by ∆(G). For X ⊆ V (G) the subgraph induced by X is denoted by G[X],
and the set of vertices in V (G) − X which are adjacent to a vertex in X is denoted by
Γ(X). We say that an edge e enters X if exactly one end-vertex of e is contained in X.
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For a subgraph P of G let G−P = G[V (G)−V (P )]. Finally, we say that an F-packing P
of G misses a vertex set X ⊆ V (G)ifX ∩ V (P )=∅ and that P covers X if X ⊆ V (P ).
2 The theorems
In this section we state the main theorems of the paper. The proofs are contained in
Sections 4 and 7. Till Section 8, k is a fixed positive integer.
Definition 2.1. A k-piece is a connected graph G with ∆(G)=k.
Definition 2.2. For a graph G we denote I
G
= G[{v ∈ V (G): deg
G
(v) ≥ k}].
Definition 2.3. AgraphG is hypomatchable if G − v has a perfect matching for all
v ∈ V (G).
In [5] it was revealed that galaxies play a central role in the k-piece packing problem.
Definition 2.4. [5] For an integer k ≥ 1 the connected graph H is a k-galaxy if it satisfies
the following properties:
• each component of I
H
is a hypomatchable graph,
• for each v ∈ V (I
H
) there exist exactly k − 1 edges between v and V (H) − V (I
H
),
each being a cut edge in H.
A hypomatchable graph has no vertex of degree 1 so a k-galaxy has no vertex of

degree k. Furthermore, each component of I
H
is a hypomatchable graph on at least 3
vertices. Since k is fixed, we shall call a k-galaxy simply a galaxy. Galaxies generalize
hypomatchable graphs because the 1-galaxies are exactly the hypomatchable graphs. The
2-galaxies were introduced by Kaneko under the name ‘sun’ [7]. See Fig. 1 for some
galaxies. The vertices of I
H
are drawn as big dots and the edges of I
H
as thick lines.
a 4-galaxy
2-galaxies
tips:
a 1-galaxy
I
H
:
Fig. 1. Galaxies
The following important property of galaxies was proved in [5].
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Lemma 2.5. [5] A k-galaxy has no perfect k-piece packing.
Now we introduce special subgraphs of galaxies, called tips. Each tip is circled by a
thin line in Fig. 1 (except in the 4-galaxy of Fig. 1 where not all tips are circled).
Definition 2.6. [5] If k ≥ 2 then for a k-galaxy H the connected components of H−V (I
H
)
are called tips.Inthecasek = 1 we call each vertex of H a tip. The union of vertex sets
ofthetipsisdenotedbyW
H

⊆ V (H).
So W
H
= V (H)ifk =1andW
H
= V (H) − V (I
H
)ifk ≥ 2. In the case k ≥ 2a
k-galaxy may consist of only a single tip (a graph with highest degree at most k − 1), but
must always contain at least one tip.
The Edmonds-Gallai structure theorem can be formulated for the k-piece packing
problem as follows. The classical Edmonds-Gallai theorem first defines the vertex set D
to consist of those vertices which can be missed by a maximal matching. In the k-piece
packing problem we have to use a different formulation. This causes the fact that Theorem
2.8 is not a direct generalization of the classical Edmonds-Gallai theorem.
Definition 2.7. For a graph G let
U
G
= {v ∈ V (G) : there exists a maximal k-piece packing P of G with v/∈ V (P ) }.
Theorem 2.8. For a graph G let D = {v : |U
G−v
| < |U
G
|}, A =Γ(D) and C = V (G) −
(D ∪ A).Now
1. the connected components of G[D] are k-galaxies,
2. for all ∅= A

⊆ A the number of those k-galaxy components of G[D] which are
adjacent to A


is at least k|A

| +1,
3. G[C] has a perfect k-piece packing,
4. a k-piece packing P of G is maximal if and only if
(a) exactly k|A| connected components of G[D] are entered by an edge of P and
these components are completely covered by P ,
(b) if H is a component of G[D] not entered by P then P[H] is a maximal k-piece
packing of H,
(c) P [C] is a perfect k-piece packing of G[C],
5. for each maximal k-piece packing P of G, the graph G−P has exactly c(G[D])−k|A|
connected components.
For proof, see Section 4. We could also choose D = {v : U
G−v
 U
G
} by Theorem
4.19.
It is a well known fact in matching theory that those vertex sets which can be covered
by a matching form a matroid. In the k-piece packing problem this property holds only
in the following weaker form. The proof is contained in Section 7.
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Theorem 2.9. There exists a partition π on V (G) and a matroid M on π such that
the vertex sets of the maximal k-piece packings are exactly the vertex sets of the form

{X : X ∈ π

} where π


isabaseofM.
3 Preliminaries
In this section we summarize the results and notions of [5] which are needed to prove the
main theorems of the paper. First we introduce two other classes of graphs which are
near to galaxies.
Definition 3.1. For an integer k ≥ 2 the connected graph H is an almost k-galaxy of
type 1 if it satisfies the following properties:
• one of the components of I
H
has a perfect matching and the others are hypomatch-
able,
• for each v ∈ V (I
H
) there exist exactly k − 1 edges between v and V (H) − V (I
H
),
each being a cut edge in H.
Definition 3.2. For an integer k ≥ 2 the connected graph H is an almost k-galaxy of
type 2 if it satisfies the following properties:
• each component of I
H
is a hypomatchable graph,
• there is a distinguished vertex w ∈ V (I
H
) such that for each v ∈ V (I
H
)eachedge
between v and V (H) − V (I
H
)isacutedgeinH, and the number of these edges is

k − 1 for v = w and k − 2 for w.
w
almost k-galaxy of type 2almost k-galaxy of type 1
Fig. 2. Almost galaxies, k =4
Fig. 2 shows some almost 4-galaxies. Just like in the case of galaxies, we define tips
for almost galaxies. Some tips are circled by a thin line in Fig. 2.
Definition 3.3. For an almost galaxy H the connected components of H − I
H
are called
tips.
Many properties of the galaxies are explained by the following lemma, which is implicit
in [5].
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Lemma 3.4. Each almost k-galaxy has a perfect k-piece packing.
Proof. First we prove the statement for almost galaxies of type 2. Let H be an almost
k-galaxy of type 2. We proceed by induction on |V (H)|.LetK be the component of I
H
containing the specified vertex w. K is a hypomatchable graph on at least 3 vertices so
it is easy to see that w has two neighbors w

,w

∈ V (K) such that K −{w

,w,w

} has
a perfect matching M. For each edge uv ∈ M let P
uv
be the subgraph of H induced by

the vertex set
{u, v}∪

{V (T ): T is a tip of H adjacent to {u, v}}.
Furthermore, let P
w
be the subgraph of H induced by the vertex set
{w

,w,w

}∪

{V (T ): T is a tip of H adjacent to {w

,w,w

}} ,
with the deletion of the edge w

w

(if any). Clearly P
uv
(uv ∈ M)andP
w
are disjoint
k-piece subgraphs of H. Deleting these k-pieces from H, each connected component of
the remaining graph is an almost k-galaxy of type 2 so we are done by induction.
Now let H be an almost k-galaxy of type 1. Denote by K the perfectly matchable

component of I
H
. For each edge uv of a perfect matching of K let P
uv
be the k-piece
subgraph of H induced by the vertex set
{u, v}∪

{V (T ): T is a tip of H adjacent to {u, v}}.
Deleting these k-pieces from H, each connected component of the remaining graph is an
almost k-galaxy of type 2 so we are done by the first part of the proof.
Lemma 3.5. [5] If T is a tip of a k-galaxy H then H − T has a perfect k-piece packing.
Proof. The statement holds for k = 1 by definition. Let k ≥ 2. It is easy to see that each
component of H − T is an almost k-galaxy of type 2, which has a perfect k-piece packing
by Lemma 3.4.
For the proof of the following lemma see [5].
Lemma 3.6. [5] If P is a k-piece packing of the k-galaxy H then there exists a tip T of
H such that V (P) ∩ V (T )=∅.
The maximal matchings of a hypomatchable graph H are exactly the perfect matchings
of H −v for the vertices v ∈ V (H). The characterization of the maximal k-piece packings
of a k-galaxy can be stated by means of the tips.
Lemma 3.7. [5] The maximal k-piece packings of a k-galaxy H are exactly the perfect
k-piece packings of H − T where T is a tip of H.
Proof. By Lemmas 3.5 and 3.6.
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The next lemma is another generalization of the defining property 2.3 of hypomatch-
able graphs. This lemma is only implicit in [5].
Lemma 3.8. If H is a k-galaxy and v ∈ V (H) then there exists a vertex set v ∈ X ⊆
V (H) such that H[X] is connected, ∆(H[X]) ≤ k − 1 and H − X has a perfect k-piece
packing.

Proof. The statement is trivial for k = 1 so assume that k ≥ 2. If v is contained in a tip
T then let X = V (T ). Now H − X has a perfect k-piece packing by Lemma 3.5 so we are
done. If v ∈ V (I
H
)thenlet
X = {v}∪

{V (T ): T is a tip of H adjacent to v}.
Clearly ∆(H[X]) = k − 1. It is easy to check that each component of H − X is an almost
k-galaxy of type 1 or 2. Hence H − X has a perfect k-piece packing by Lemma 3.4.
Definition 3.9. A connected graph G is a k-solar-system (see Fig. 3)ifithasavertex
y, called center, such that deg
G
(y)=k and G − y has k connected components, each
being a k-galaxy.
.
.
.
.
k-galaxies
y
v
1
H
2
v
k
H
k
H

1
v
2
Fig. 3. A k-solar system
Lemma 3.10. Each k-solar-system has a perfect k-piece packing.
Proof. Let G be a k-solar-system with center y. Denote the neighbors of y by v
i
(1 ≤ i ≤
k) and denote the k-galaxy component of G − y containing v
i
by H
i
. Lemma 3.8 implies
that for all 1 ≤ i ≤ k there exists a vertex set v
i
∈ X
i
⊆ V (H
i
) such that H
i
− X
i
has
a perfect k-piece packing and H
i
[X
i
] is a connected graph with highest degree at most
k − 1. The latter condition on H

i
[X
i
] implies that G[{y}∪

1≤i≤k
X
i
]isak-piece.
[5] describes a polynomial algorithm finding a maximum k-piece packing in the input
graph G. The algorithm consists of two phases and already the first phase obtains a max-
imal k-piece packing of G which is further refined in the second phase (called ’Re-Rooting
procedure’) to become a maximum k-piece packing. Now we are interested only in the
first phase of the algorithm of [5] to which we simply refer as the algorithm. This algo-
rithm is a direct generalization of the alternating forest matching algorithm of Edmonds.
the electronic journal of combinatorics 12 (2005), #R8 7
It builds certain alternating forests and it outputs a decomposition V (G)=D ∪ A ∪ C
where the sets D, A, C are pairwise disjoint. It also outputs a maximal k-piece packing
P of G but we are not interested in it now. The algorithm may have different runs on
the same graph G depending on the actual implementation. We refer to the outputs of
these runs as decomposition outputs. In the next section we prove that the decomposition
output is unique for all runs of the algorithm and it is canonical for the k-piece packing
problem in a certain way. The following proposition is implicit in the description of the
algorithm in [5], see Fig. 4.
Proposition 3.11. [5] Each run of the algorithm outputs a decomposition V (G)=D ∪
A ∪ C where D, A, C are pairwise disjoint and
1. the connected components of G[D] are k-galaxies,
2. G contains no edge joining D to C,
3. for all ∅= A


⊆ A the number of those k-galaxy components of G[D] which are
adjacent to A

is at least k|A

| +1,
4. G[C] has a perfect k-piece packing.
A:
k-galaxy components
D:
C:
G[C] has a perfect k-piece packing
Fig. 4. A decomposition output of the algorithm, k =2
Any decomposition output of the algorithm implies the Tutte-type existence theorem
3.13 for the k-piece packing problem, proved in [5].
Definition 3.12. Let k-gal(G) denote the number of those connected components of the
graph G that are k-galaxies.
Theorem 3.13. [5] A graph G has a perfect k-piece packing if and only if
k-gal(G − A) ≤ k|A|
for all set of vertices A ⊆ V (G).
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Proof. The “only if” part is straightforward using that a k-galaxy has no k-piece packing
by Lemma 2.5. On the other hand, if G has no perfect k-piece packing then A in any
decomposition output of the algorithm will do.
4 The Edmonds-Gallai decomposition
In this section we prove that the decomposition output is unique for all runs of the
algorithm and that this decomposition has the properties described in Theorem 2.8.
Definition 4.1. For A ⊆ V (G)let
D
A

=

{V (H): H is a k-galaxy component of G − A}.
We use the notation D
A
G
if confusion may arise. Moreover, let C
A
= V (G) − (D
A
∪ A)(or
C
A
G
).
Definition 4.2. The vertex set A ⊆ V (G)hask-surplus if for all ∅= A

⊆ A the number
of k-galaxy components of G[D
A
] adjacent to A

is at least k|A

| + 1. The vertex set A is
perfect if C
A
has a perfect k-piece packing.
Definition 4.3. WesaythatavertexsetA ⊆ V (G)canbek-matched into X ⊆ V (G)−A
by M if M is a subgraph of G with k|A| edges such that deg

M
(v)=k for all v ∈ A and
exactly k|A| connected components of G[X] are entered by an edge of M (each by one
edge). The vertex set A can be k-matched into X ⊆ V (G) − A if there exists a subgraph
M of G such that A can be k-matched into X by M.
The following property (in fact, characterization) of the vertex sets with k-surplus is
implied by Hall’s theorem.
Lemma 4.4. If A ⊆ V (G) has k-surplus then A can be k-matched into D
A
− V (H) for
each connected component H of G[D
A
].
Using these definitions we can reformulate Proposition 3.11.
Proposition 4.5. For any decomposition output V (G)=D ∪ A ∪ C of the algorithm the
set A is perfect with k-surplus.
Proof. A k-galaxy has no perfect k-piece packing so D
A
= D and C
A
= C. So Proposition
3.11, 3. is tantamount to that A has k-surplus and 4. to that A is perfect.
The next lemma describes an important property of the galaxies.
Lemma 4.6. If H is a k-galaxy and ∅= X ⊆ V (H) then k-gal(H − X) ≤ k|X|−1.
Proof. The statement is well-known for k = 1. Indeed, otherwise for x ∈ X the number
of hypomatchable components of (H − x) − (X − x)ismorethan|X − x| implying that
H − x has no perfect matching, a contradiction.
For k ≥ 2 it is easier to prove the lemma for a broader set of graphs, called pseudo
galaxies.
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Definition. For an integer k ≥ 2 the connected graph G is a pseudo k-galaxy if for each
v ∈ V (I
G
) there exist exactly k − 1 edges between v and V (G) − V (I
G
),eachbeingacut
edge in G.
Note, that this is just the definition of the k-galaxies with the relaxation that the
connected components of I
G
need not be hypomatchable. What we actually prove is
Lemma 4.7 which immediately implies Lemma 4.6.
Lemma 4.7. If G is a pseudo k-galaxy and ∅= X ⊆ V (G) is a vertex set with the
property that each vertex of X ∩ V (I
G
) is contained in a hypomatchable component of I
G
then k-gal(G − X) ≤ k|X|−1 holds.
Proof. Suppose that G is a pseudo galaxy of minimum size for which a vertex set ∅=
X ⊆ V (G) fails Lemma 4.7, ie. k-gal(G − X) ≥ k|X| holds. deg
G
(v) ≤ k − 1 for vertices
v/∈ V (I
G
) so clearly X ∩ V (I
G
) = ∅.
Let F be a hypomatchable component of I
G
with X

F
= X ∩ V (F ) = ∅. Assume that
the number of k-galaxy components of G − X
F
is s and denote these components by
H
1
, ,H
s
. It is easy to see that the other components of G − X
F
are pseudo k-galaxies.
Let their number be t and denote them by G
1
, ,G
t
. Note that each component K of
G − X
F
satisfies the condition of Lemma 4.7, ie. each vertex of (X ∩ V (K)) ∩ V (I
K
)is
contained in a hypomatchable component of I
K
.Leth (resp. g) denote the number of
vertices x ∈ X contained in a k-galaxy (resp. pseudo k-galaxy) component of G − X
F
.
Clearly |X| = |X
F

| + h + g.
Let X
i
= X ∩ G
i
for 1 ≤ i ≤ t. By induction, k-gal(G
i
− X
i
) ≤ k|X
i
| for 1 ≤ i ≤ t
independently of the emptiness of X
i
.Sothenumberofk-galaxy components of G − X
contained in a component G
i
for 1 ≤ i ≤ t is at most kg.
Now we bound s.LetH
i
be a k-galaxy component of G − X
F
such that Y = V (H
i
) ∩
V (F ) = ∅. It is easy to see that F[Y ] is connected. This implies that F [Y ] is a component
of I
H
i
so it is hypomatchable. The number of such hypomatchable components F [Y ]is

at most k|X
F
|−1 by the already proved case k = 1 of Lemma 4.6. Thus the number
of k-galaxy components of G − X
F
which intersect V (F )isatmostk|X
F
|−1. On the
other hand, the number of components of G − X
F
which do not intersect V (F )isexactly
(k − 1)|X
F
| because each vertex v ∈ X
F
⊆ V (F ) is incident with exactly k − 1cutedges
in G.Sos ≤|X
F
|−1+(k − 1)|X
F
| = k|X
F
|−1.
Let s

be the number of those k-galaxy components H
i
of G − X
F
for which X

i
=
X ∩V (H
i
) = ∅. For such a component k-gal(H
i
−X
i
) ≤ k|X
i
|−1 holds by the minimality
of G. So these components contain altogether at most kh−s

of the k-galaxy components
of G − X. Finally, it is trivial that the number of k-galaxy components H
i
of G − X
F
for
which X ∩ V (H
i
)=∅ is s − s

. Summarizing,
k-gal(G − X) ≤ kg +(kh− s

)+(s − s

) ≤ k(h + g)+s ≤ k(|X
F

| + h + g) − 1=k|X|−1.
Theorem 4.8. If A
1
,A
2
⊆ V (G) are perfect vertex sets with k-surplus then A
1
= A
2
.
the electronic journal of combinatorics 12 (2005), #R8 10
Proof. Let D
i
= D
A
i
and C
i
= C
A
i
for i =1, 2. Denote by g
i
the number of components
of G[D
i
] intersecting A
3−i
for i =1, 2. We prove that g
1

= g
2
= 0. Suppose that g
1
≥ g
2
and that A

2
= A
2
∩ D
1
= ∅.Bythek-surplus of A
2
, the vertex set A

2
is adjacent to at
least k|A

2
| +1 k-galaxy components of G[D
2
]. Let K be a k-galaxy component of G[D
2
]
which is adjacent to A

2

.IfV (K) ∩ A
1
= ∅ then V (K) ⊆ D
1
because A

2
⊆ D
1
so K is
contained in a k-galaxy component of G[D
1
]. Thus the number of such components K
with V (K) ∩ A
1
= ∅ is at most k|A

2
|−g
1
by Lemma 4.6. So the number of components
of G[D
2
] which are adjacent to A

2
and intersect A
1
is at least g
1

+1. Thusg
2
≥ g
1
+1, a
contradiction. This implies g
1
= g
2
=0.
Suppose that A
1
\ A
2
= ∅.Bythek-surplus of A
1
the number of components of G[D
1
]
which are adjacent to A
1
\A
2
is at least k|A
1
\A
2
|+1. These components do not intersect
A
2

because g
1
= 0. Hence k-gal(G[C
2
] − (A
1
\ A
2
)) ≥ k|A
1
\ A
2
| + 1 implying that G[C
2
]
has no perfect k-piece packing by Theorem 3.13, a contradiction.
So A
1
⊆ A
2
and by symmetry, A
1
= A
2
.
Theorem 4.9. The decomposition output is unique for all runs of the algorithm.
Proof. Let V (G)=D∪A∪ C be any decomposition output of the algorithm. Proposition
4.5 implies that A is perfect with k-surplus hence it is unique by Theorem 4.8. Finally, a
k-galaxy has no perfect k-piece packing so D = D
A

and C = C
A
.
Hence the following definition is sound:
Definition 4.10. The unique decomposition output of the algorithm is denoted by
V (G)=D
G
∪ A
G
∪ C
G
andcalledthecanonical decomposition of G with respect to
the k-piece packing problem.
Proposition 4.5 and Theorem 4.8 imply
Corollary 4.11. If A ⊆ V (G) is perfect and has k-surplus then A = A
G
.
Now we investigate the structure of maximal k-piece packings of G.
Lemma 4.12. Each maximal k-piece packing P of G has the following structure:
1. exactly k|A
G
| connected components of G[D
G
] are entered by an edge of P and these
components are completely covered by P ,
2. if H is a component of G[D] not entered by P then P [H] is a maximal k-piece
packing of H, ie. there exists a tip T of H such that P [H] is a perfect k-piece
packing of H − T , and
3. P [C
G

] is a perfect k-piece packing of G[C
G
].
Proof. Let P be a maximal k-piece packing of G. We construct a k-piece packing P

with
V (P

) ⊇ V (P ) such that if P fails any of properties 1 3. then V (P

)  V (P ) would hold.
We need the theorem of Mendelsohn and Dulmage (see 1.4.3 in [11]).
the electronic journal of combinatorics 12 (2005), #R8 11
Theorem 4.13. (Mendelsohn, Dulmage) Let B be a bipartite graph with color classes
U and V .IfB has a matching covering U

⊆ U and another matching covering V

⊆ V
then it has a matching covering U

∪ V

.
We apply Theorem 4.13 to the bipartite graph B
A
defined as follows.
Definition 4.14. We denote kA
G
= {v

i
: v ∈ A
G
, 1 ≤ i ≤ k}.LetV (B
A
)=kA
G
∪
{H : H is a component of G[D
G
]} and E(B
A
)={v
i
H :1≤ i ≤ k, v is adjacent to H in G}.
B
A
has a matching covering kA
G
by the k-surplus of A
G
.Moreover,P shows that B
A
has a matching covering H
P
= {H : H is a component of G[D
G
] entered by an edge of P }.
So Theorem 4.13 implies that B
A

has a matching M with vertex set kA
G
∪H
M
where
H
P
⊆H
M
. Using Lemma 3.10, this matching gives rise to a perfect k-piece packing P
1
in the subgraph induced by
A
G
∪

{V (H): H ∈H
M
}.
Let H be a component of G[D
G
] such that H/∈H
M
. By Lemma 3.6 there exists a tip
T of H such that V (P) ∩ V (T )=∅. Take a perfect k-piece packing of H − T guaranteed
by Lemma 3.5 and denote the union of these k-pieces by P
2
. Finally, let P
3
be a perfect

k-piece packing of G[C
G
]. With P

= P
1
∪ P
2
∪ P
3
we get that V (P

) ⊇ V (P ).
Trivially |H
P
|≤k|A
G
|. In fact, |H
P
| = k|A
G
| holds here because otherwise the
matching M of B
A
would enter strictly more components of G[D
G
]thanP , resulting
in V (P

)  V (P ), a contradiction. Properties 1. and 2. are straightforward by the

maximality of P and by Lemmas 3.7 and 3.10. For 3. observe that P has no edge joining
A
G
to C
G
because otherwise |H
P
| <k|A
G
| would hold.
Observe that Lemma 4.12 holds also by replacing A
G
by A, D
G
by D
A
and C
G
by C
A
where A ⊆ V (G) is a perfect vertex set which can be k-matched into D
A
. This observation
will be needed in the proof of Theorem 4.19.
Lemma 4.15. If P is a k-piece packing satisfying properties 1., 2. and 3. of Lemma 4.12
then P is maximal.
Proof. Properties 1., 2. and 3. imply that c(G − P )=c(G[D
G
]) − k|A
G

| and that each
component of G − P is a tip of some galaxy component of G[D
G
]. Let H
P
= {H : H is
a component of G[D
G
] entered by an edge of P }. Suppose that P

is a k-piece packing
covering V (P ) and one more vertex v/∈ V (P). Now v is contained in a tip of a galaxy
H/∈H
P
. So Property 2. implies that P

intersects each tip of H thus P

enters H by
Lemma 3.6. Moreover, P

enters each component in H
P
by Lemma 3.6. So P

enters
at least k|A
G
| + 1 components of G[D
G

] which is impossible because deg
P

(v) ≤ k for
v ∈ A
G
.
For characterizing D
G
in the canonical decomposition first we need to characterize the
union of the vertex sets of tips in G[D
G
]. Recall that U
G
was introduced in Definition 2.7.
Definition 4.16. Let W
G
=

{W
H
: H is a k-galaxy component of G[D
G
]}.
the electronic journal of combinatorics 12 (2005), #R8 12
Lemma 4.17. W
G
= U
G
.

Proof. Lemma 4.12 implies that U
G
⊆ W
G
. On the other hand, let v ∈ W
G
be a vertex
contained in a tip T of a k-galaxy component H
0
of G[D
G
]. A
G
has k-surplus so A
G
can
be k-matched into D
G
−V (H
0
) by a subgraph M of G.LetH
M
= {H : H is a component
of G[D
G
] entered by an edge of M}. Using Lemma 3.10, M gives rise to a perfect k-piece
packing P
1
in the subgraph induced by A
G

∪

{V (H): H ∈H
M
}. By Lemma 3.7, for
each component H/∈H
M
of G[D
G
] we can take a perfect k-piece packing of H −T
H
where
T
H
is any tip of H. Take care to choose T
H
0
= T . The union of these k-pieces is denoted
by P
2
. Finally, let P
3
be a perfect k-piece packing of G[C
G
]. By Lemma 4.15, the k-piece
packing P
1
∪ P
2
∪ P

3
is maximal and it misses v ∈ W
G
.
In the matching case (ie. in the case k = 1) it holds that W
G
= D
G
thus Lemma 4.17
itself characterizes the canonical D
G
. In the general case only W
G
⊆ D
G
holds so we have
to go one step further in order to characterize D
G
in Theorem 4.19. First we need the
following lemma.
Lemma 4.18. If H is a k-galaxy and v ∈ V (H) then each component of H − v is either
a k-galaxy or has a perfect k-piece packing. Moreover,

{W
K
: K is a k-galaxy component of H − v}  W
H
.
Proof. The statement is well-known for k = 1 so assume k ≥ 2. If v is contained in a
tip then clearly each component of H − v is either a k-galaxy or an almost k-galaxy of

type 2. Each almost k-galaxy component has a perfect k-piece packing by Lemma 3.4.
Furthermore,

{V (T): T is a tip in a component of H − v} = W
H
− v
so we are done. If v ∈ I
H
then H − v consists of k-galaxy components (the number of
which is exactly k − 1), and almost galaxy components of type 1, the number of which is
at least 1. Each almost k-galaxy component has a perfect k-piece packing by Lemma 3.4.
Moreover,

{V (T): T is a tip in a component of H − v} = W
H
,
but each almost galaxy component contains at least one tip of H, yielding that

{V (T): T is a tip in an almost k-galaxy component of H − v}= ∅.
Theorem 4.19. D
G
= {v : U
G−v
 U
G
} = {v : |U
G−v
| < |U
G
|} holds for all graphs G.

Proof. We investigate the canonical decomposition of the graph G − v.
1. Let v ∈ C
G
. Denote the graph G[C
G
− v]byG

. Observe that in the graph G − v
the set A
G
∪ A
G

is perfect with k-surplus. So A
G−v
= A
G
∪ A
G

by Corollary 4.11,
yielding that W
G−v
⊇ W
G
,ie.U
G−v
⊇ U
G
by Lemma 4.17.

the electronic journal of combinatorics 12 (2005), #R8 13
2. Let v ∈ A
G
. In the graph G − v the set A
G
− v is perfect with k-surplus so
A
G−v
= A
G
− v by Corollary 4.11. Hence W
G−v
= W
G
or equivalently, U
G−v
= U
G
by Lemma 4.17.
3. Finally, suppose that v ∈ V (H) for a k-galaxy component H of G[D
G
]. ∅ is perfect
and has k-surplus in the graph H − v by Lemma 4.18 so A
H−v
= ∅ by Corollary
4.11, yielding that
D
H−v
= {V (K): K is a k-galaxy component of H − v} and
C

H−v
= {V (K): K is a component of H − v with a perfect k-piece packing}.
Let D

= D
G−v
A
G
=(D
G
\ V (H)) ∪ D
H−v
, C

= C
G−v
A
G
= C
G
∪ C
H−v
and W

=
{V (T): T is a tip in a component of G[D

]}. Lemma 4.18 implies that W

 W

G
.
In the graph G−v the set A
G
is perfect because G[C

] has a perfect k-piece packing.
Moreover, A
G
can be k-matched into D

in G − v because A
G
has k-surplus in G.
So the statement of Lemma 4.12 holds for A
G
in the graph G − v, as we mentioned
after the proof of 4.12. This especially implies that each maximal k-piece packing
of G − v misses only vertices in W

.SoU
G−v
⊆ W

 W
G
= U
G
and we are done.
At this point the proof of Theorem 2.8 is straightforward using the results of this

section.
Proof of Theorem 2.8. D = D
G
,A= A
G
and C = C
G
by Theorem 4.19. Now Property
1. holds by definition. A
G
is perfect with k-surplus which is just tantamount to Properties
2. and 3. Property 4. is equivalent to Lemmas 4.12 and 4.15. Finally, 5. follows from
Property 4.
By Theorem 2.8 the graph G has a canonical decomposition V (G)=D
k
∪ A
k
∪ C
k
for
each k ≥ 1. Here D
1
∪ A
1
∪ C
1
is the classical Edmonds-Gallai decomposition. Observe
that A
k
= C

k
= ∅ if k ≥ ∆(G)+1andD
k
= A
k
= ∅ if k =∆(G). Nevertheless, there
does not seem to be any nice relation between the decompositions for different k’s.
5 The calculus of barriers
In this section we prove some properties of barriers which we define to be those vertex
sets A which maximize k-gal(G − A) − k|A|. Not all of the following results generalize
the theory of barriers described by Lov´asz and Plummer [11] because they count the odd
size components instead of the hypomatchable components as we do.
Definition 5.1. For A ⊆ V (G)thedeficiency of A is def(A)=k-gal(G − A) − k|A|.The
deficiency of G is
def(G)=max{def(A): A ⊆ V (G)}.
Finally, A ⊆ V (G)isabarrier if def(A)=def(G).
the electronic journal of combinatorics 12 (2005), #R8 14
Theorem 3.13 is tantamount to saying that G has a perfect k-piece packing if and only
if def(G)=0. Inthiscase∅ is a barrier with deficiency 0.
Proposition 5.2. A
G
is a barrier of G.
Proof. Let P be a maximal k-piece packing of G. Lemma 4.12 implies that c(G − P )=
k-gal(G−A
G
)−k|A
G
| =def(A
G
). On the other hand, let A be a barrier of G.Thenumber

of components of G[D
A
] which are not entered by P is clearly at least k-gal(G−A)−k|A| =
def(A). Thus c(G − P ) ≥ def(A) by Lemma 2.5. This implies that def(A
G
) ≥ def(A)and
so that A
G
is a barrier.
In the matching case (ie. when k = 1) each maximum (and so each maximal) matching
misses def(G) vertices of G. This property fails for general k because a maximal k-piece
packing of a galaxy may miss an arbitrary number of vertices instead of only one (namely,
the vertices of a tip). What is salvaged, is that c(G − P )=def(G) for each maximal
k-piece packing P by Lemma 4.12 and Proposition 5.2.
Lemma 5.3. Each barrier is perfect.
Proof. Let A be a barrier of G. Assume that G[C
A
] has no perfect k-piece packing. Then
by Theorem 3.13 there exists a set X ⊆ C
A
such that k-gal(G[C
A
] − X) − k|X| > 0. But
then def(A ∪ X) > def(G) would hold, a contradiction.
Theorem 5.4. If A is a barrier then A
G
⊆ A and D
G
⊆ D
A

.
Proof. Let A be a barrier of G and let H = {H : H is a component of G[D
A
]}. For J⊆H
let
Γ(J )=

v ∈ A : v is adjacent to

{V (H): H ∈J}

.
Consider the following function f on H: for J⊆Hlet f(J )=|J | − k|Γ(J )|. Clearly
f(J ) ≤ def(G) for J⊆Hand f is a supermodular function. Suppose that f (J
1
)=
f(J
2
)=def(G) for J
1
, J
2
⊆H.Now2·def(G)=f(J
1
)+f(J
2
) ≤ f(J
1
∩J
2

)+f(J
1
∪J
2
) ≤
2 · def(G) implying that f (J
1
∩J
2
)=def(G). f(H)=def(G) thus there exists an
inclusion-wise minimum set H
0
⊆Hwith f(H
0
)=def(G). Let A
0
=Γ(H
0
). The set A
0
has k-surplus because H
0
is minimum.
Let D

=

{V (H): H ∈H−H
0
}. We state that A − A

0
can be k-matched into D

by a subgraph M of G. This is due to Hall’s theorem: if Y ⊆ A − A
0
was adjacent to less
than k|Y | components of G[D

]thendef(A − Y ) > def(A)=def(G) would hold because
Y is not adjacent to any component H ∈H
0
.Moreover,k|A − A
0
| = |H − H
0
| so M
gives rise to a perfect k-piece packing in D

∪ (A − A
0
) using Lemma 3.10. Moreover, by
Lemma 5.3, G[C
A
] has a perfect k-piece packing so A
0
is perfect.
Summarizing, A
0
is perfect with k-surplus so A
G

= A
0
⊆ A by Corollary 4.11. More-
over, clearly D
G
= D
A
0
= D
A
− D

.
Note that in this proof, A − A
0
is adjacent to at most k|A − A
0
| components in G[D
A
]
hence if A has k-surplus then A − A
0
= ∅. This implies that A
G
is the only barrier with
k-surplus.
the electronic journal of combinatorics 12 (2005), #R8 15
Theorem 5.5. The intersection of two barriers is a barrier.
Proof. Let A
1

,A
2
be barriers of G.WeletD
i
= D
A
i
and C
i
= C
A
i
for i =1, 2. Denote
by g
i
the number of components of G[D
i
] intersecting A
3−i
. Wlog. we may assume that
g
1
≤ g
2
. Furthermore,
• g
C
is the number of components of G[D
1
] contained in C

2
,
• g
D
is the number of components of G[D
1
] contained in D
2
and not adjacent to
A
1
∩ D
2
,
• g

D
is the number of components of G[D
1
] contained in D
2
and adjacent to A
1
∩ D
2
.
Now
k|A
1
| +def(G)=k-gal(G − A

1
)=g
C
+ g
1
+ g
D
+ g

D
.
The graph G[C
2
] has a perfect k-piece packing by Lemma 5.3 so
g
C
≤ k|A
1
∩ C
2
|.
The components of G[D
1
] which are contained in D
2
but which are not adjacent to A
1
∩D
2
are connected components of G − (A

1
∩ A
2
)aswellso
g
D
≤ k-gal(G − (A
1
∩ A
2
)).
Each component of G[D
1
] which is contained in D
2
and which is adjacent to A
1
∩ D
2
is contained in some component H of G[D
2
]. The number of such components H was
denoted by g
2
. Hence Lemma 4.6 implies that
g

D
≤ k|A
1

∩ D
2
|−g
2
.
Summarizing,
k|A
1
| +def(G) ≤ k| A
1
∩ C
2
| + k|A
1
∩ D
2
| + g
1
− g
2
+ k-gal(G − (A
1
∩ A
2
)) ≤
≤ k|A
1
| + k-gal(G − (A
1
∩ A

2
)) − k|A
1
∩ A
2
|.
So def(G) ≤ def(A
1
∩ A
2
), ie. A
1
∩ A
2
is a barrier.
Theorem 5.6. If A
1
and A
2
are barriers such that there is no edge between A
1
∩ D
A
2
and A
2
∩ D
A
1
then A

1
∪ A
2
is a barrier.
Proof. Let D
i
= D
A
i
and C
i
= C
A
i
for i =1, 2. We prove that A
1
∩ D
2
and A
2
∩ D
1
are empty. Assume that A
1
∩ D
2
= ∅ and let K be a component of G[D
2
] such that
X = A

1
∩ V (K) = ∅ . X ⊆ A
1
is adjacent to at least k|X| components of G[D
1
]since
otherwise def(A
1
− X) > def(G) would hold. Let v ∈ D
1
be a vertex adjacent to x ∈ X.
v/∈ C
2
since G contains no edge between D
2
and C
2
. v/∈ A
2
either by the condition of
the theorem. Hence v is contained in the same component of G[D
2
]thanx,ie.v ∈ V (K).
But then Lemma 4.6 implies that X can have at most k|X|−1 neighbors among the
components of G[D
1
], a contradiction.
So A
1
∩ D

2
= ∅ and by symmetry A
2
∩ D
1
= ∅.Let
the electronic journal of combinatorics 12 (2005), #R8 16
• g
1
C
be the number of components of G[D
1
] contained in C
2
,
• g
2
C
be the number of components of G[D
2
] contained in C
1
and
• g
D
= c(G[D
1
∩ D
2
]).

Clearly
k ·|A
1
| +def(G)=k-gal(G − A
1
)=g
1
C
+ g
D
,
k ·|A
2
| +def(G)=k-gal(G − A
2
)=g
2
C
+ g
D
and
k ·|A
1
∩ A
2
| +def(G)=k-gal(G − (A
1
∩ A
2
)) ≥ g

D
.
These inequalities sum up to g
D
+ g
1
C
+ g
2
C
≥ k ·|A
1
∪ A
2
| +def(G). It is easy to see that
k-gal(G − (A
1
∪ A
2
)) ≥ g
D
+ g
1
C
+ g
2
C
and so A
1
∪ A

2
is a barrier.
Theorem 5.6 fails for arbitrary barriers. For example, let k =2andP
3
be the path
of length 3 with vertices v
1
,v
2
,v
3
,v
4
in this order. P
3
has a perfect 2-piece packing so
C
P
3
= V (P
3
). The barriers of P
3
are A
P
3
= ∅, {v
2
} and {v
3

} but {v
2
,v
3
} is not a barrier.
In the matching theory, the deficiency is usually defined as q(G − A) −|A| where
q(G − A) is the number of odd size components of G − A. For this ’odd-deficiency’ it
holds that A
G
∪ C
G
is the union of inclusion-wise maximal barriers. This property fails
for our deficiency, see P
3
defined in the previous paragraph.
For the odd-deficiency it also holds that A
G
is the intersection of the inclusion-wise
maximal barriers. This property fails in our case as well. For example, let P
2
be the path
of length 2 with vertices v
1
,v
2
,v
3
in this order. P
2
has a perfect 2-piece packing and its

barriers are A
P
2
= ∅ and {v
2
}.
Nevertheless, Theorem 5.5 fails for the classical odd deficiency.
6 Two more properties of galaxies
First we show a characterization of k-galaxies which is a direct generalization of the
defining property 2.3 of the hypomatchable graphs.
Theorem 6.1. A graph G satisfies properties 1. and 2. if and only if G is a k-galaxy.
1. G has no perfect k-piece packing.
2. For each v ∈ V (G) there exists a vertex set v ∈ X ⊆ V (G) such that G[X] is
connected, ∆(G[X]) ≤ k − 1 and G − X has a perfect k-piece packing.
Proof. If G is a k-galaxy then 1. follows from Lemma 2.5 and 2. from Lemma 3.8.
For the reverse direction, suppose that G satisfies the above two properties. First, if
A
G
= ∅ then either C
G
= V (G) which contradicts to 1. by Theorem 2.8 property 3.,or
D
G
= V (G). In this latter case each component of G is a k-galaxy. However, G cannot
have more than one component since then 2. would yield a perfect k-piece packing of G
contradicting to 1. Second, assume that A
G
= ∅. Choose a vertex v ∈ A
G
and let X be

the electronic journal of combinatorics 12 (2005), #R8 17
the vertex set guaranteed by 2. Now deg
G[X]
(v) ≤ k − 1since∆(G[X]) ≤ k − 1. Adjoin
k − deg
G[X]
(v) new isolated vertices to G and join each new vertex to v by an edge. The
new graph is denoted by G

.NowX and the set of new vertices induce a k-piece in G

.
This k-piece together with the perfect k-piece packing of G − X gives a perfect k-piece
packing of G

. However, k-gal(G

− A
G
) ≥ k|A
G
| + 1 by Theorem 2.8, property 2.,which
is a contradiction by Theorem 3.13.
In the case k =1Theorem6.12. is equivalent to the defining property 2.3 of hy-
pomatchable graphs. This implies property 1. as well by parity arguments when k =1.
However, parity has no consequence in the case k ≥ 2. Another easy characterization of
galaxies is the following corollary of Theorem 4.19.
Proposition 6.2. The following statements are equivalent for a connected graph G.
1. G is a k-galaxy.
2. |U

G−v
| < |U
G
| for all v ∈ V (G).
3. U
G−v
 U
G
for all v ∈ V (G).
Proof. 1. ⇒2. and 1. ⇒3.: ∅ is a perfect set with k-surplus so A
G
= ∅ by Corollary 4.11.
So D
G
= V (G)andboth2. and 3. are implied by Theorem 4.19.
2. ⇒1. and 3. ⇒1.: Theorem 2.8 yields that D
G
= V (G) hence G is a k-galaxy by
Theorem 2.8 property 1. and by the connectivity of G.
7 The matroidal property and maximum packings
Definition 7.1. We say that the F-packing problem is matroidal if for all graphs G those
vertex sets X ⊆ V (G) which can be covered by an F-packing of G form a matroid.
Loebl and Poljak conjecture [9] that for graph sets F with K
2
∈Fthe F-packing
problem is polynomial if and only if it is matroidal. This conjecture is still open. In [5] it
was shown that the k-piece packing problem is not matroidal in the case k ≥ 2. For an
example, let k =2andG be a claw (ie. a 3-star) with one of its edges subdivided by a
new vertex. Still, the k-piece packing problem has the matroidal property in a somewhat
weaker form. So Theorem 2.9 gives another support for the validity of the conjecture of

Loebl and Poljak.
Theorem. 2.9. There exists a partition π on V (G) and a matroid M on π such that
the vertex sets of the maximal k-piece packings are exactly the vertex sets of the form

{X : X ∈ π

} where π

isabaseofM.
Proof. Lemmas 4.12, 4.15 and the k-surplus of A
G
imply that the following considerations
hold.
π = {{v} : v/∈ W
G
}∪{V (T ): T is a tip of a k-galaxy component of G[D
G
]} .
the electronic journal of combinatorics 12 (2005), #R8 18
Denote by N the matroid with ground set H = {H : H is a component of G[D
G
]} such
that a set H

⊆Hof size k|A
G
| is a base in N if and only if A
G
can be k-matched into


{V (H): H ∈H

}.ObservethatN is indeed a matroid, it is the transversal matroid of
the bipartite graph B
A
, see Definition 4.14. Now for each component H of G[D
G
] replace
H in N by T
H
= {V (T ): T is a tip of H}⊆π such that the elements of T
H
are in series
with each other. The resulting matroid is N

with ground set {V (T ): T is a tip of a
k-galaxy component of G[D
G
]}⊆π. Add as a direct sum to N

the elements {v} as a
bridge for v/∈ W
G
. The resulting matroid is M.
The co-rank of N and N

are def(G)thustheco-rankofM is def(G)too.Notethat
for each maximal k-piece packing P of G, every vertex set of π is either fully covered or
fully missed by P and the number of the fully missed sets is def(G). In the case k =1a
tip has exactly one element so π is the partition into singletons. In the case k =2atip

has one or two elements so the vertex sets of π are of size one or two. Finally, for k ≥ 3
a tip may be of arbitrary size thus a vertex set of π can be of arbitrary size as well.
Because the ground set of the matroid M is a partition into different size sets, in the
k-piece packing problem a maximal packing is not necessarily maximum,asitisthecasein
the polynomial packing problems with K
2
∈F. Still, the vertex sets which can be covered
by maximum k-piece packings admit a similar matroid: take the maximum weight bases
of M with the weight function X → | X| for X ∈ π. This weighted matroidal approach
yields a proof for the Berge-type formula of [5] on the size of a maximum k-piece packing.
Indeed, the maximum weight bases of M correspond to the minimum weight bases of N
(defined in the proof of Theorem 2.9) with the weight function H → (the minimum size of
atipofH). So one can apply the greedy method to the k-galaxy components of G[D
G
].
In fact, a little additional work is needed for proving Theorem 7.2 since it is stated in a
more compact form in [5]. Let k-gal
i
(G) denote the number of k-galaxy components H
of the graph G with the property that each tip of H has size at least i.
Theorem 7.2. [5] If G is a graph of size n then the size of the maximum k-piece packings
of G is
n − max
n

i=1
(k-gal
i
(G − A
i

) − k|A
i
|) ,
taken over all sequences of vertex sets V (G) ⊇ A
1
⊇ A
2
⊇ ⊇ A
n
.
A
1
can be chosen to be the canonical barrier A
G
. The sequence of vertex sets is related
to the structure of the minimum weight bases of the transversal matroid N .Wedonot
go into details. In the case k = 1 we get the Berge-Tutte theorem on maximum matchings
[1]. The case k = 2 was proved by Kano, Katona and Kir´aly [8].
8The(l, u)-piece packing problem
As a generalization of the k-piece packing problem, the (l, u)-piece packing problem is
introduced in [5]. It turns out that all the above results hold with the straightforward
the electronic journal of combinatorics 12 (2005), #R8 19
modifications. We do not go into details, only illustrate this relation using the reduction
to the k-piece packing problem shown in [5].
Let two integer bounds u(v) ≥ l(v) ≥ 0 be given for each vertex v ∈ V (G). A
connected subgraph P of G is an (l, u)-piece if deg
P
(v) ≤ u(v) holds for each v ∈ V (P )
and there exists at least one vertex w ∈ V (P )withdeg
P

(w) ≥ l(w). Note that l ≡ u ≡ k
gives the k-piece packing problem. Galaxies and tips change in the following way.
Definition 8.1. Given the bounds l, u : V (H) → N, the graph H is an (l, u)-galaxy if it
satisfies the following properties:
• denoting by I
H
the graph induced by the vertices v with deg
G
(v) ≥ l(v), each
component of I
H
is a hypomatchable graph,
• l(v)=u(v) ≥ 1 for v ∈ V (I
H
),
• for each v ∈ V (I
H
) there exist exactly l(v) − 1 edges between v and V (H) − V (I
H
),
each being a cut edge in H.
The tips are the connected components of H − V (I
H
) together with the vertices v ∈
V (I
H
)withl(v)=u(v) = 1 as single vertex subgraphs.
The difference in the definition of the galaxies and tips can be explained by the follow-
ing reduction to the k-piece packing problem, described in [5]. Let k =1+max{u(v): v ∈
V (G)}. For each vertex v ∈ V (G)letM

v
and N
v
be disjoint sets of new vertices with
|M
v
| = u(v) − l(v)+1and|N
v
| = k − u(v) − 1. Now for each v ∈ V (G)takeacomplete
graph on M
v
and join the vertices of M
v
∪ N
v
to v. Denote the new graph by G
k
.Itis
easy to see that G
k
has a perfect k-piece packing if and only if G has a perfect (l, u)-piece
packing, and that G is an (l, u)-galaxy if and only if G
k
is a k-galaxy. With the help of
this reduction one can see that all the above considerations for the k-piece packings hold
for the (l, u)-piece packings as well, with the necessary modifications. For illustrating this,
we briefly describe how to get the canonical decomposition of G related to the (l, u)-piece
packing problem.
Let V (G
k

)=D
k
˙
∪ A
k
˙
∪ C
k
be the canonical decomposition of G
k
related to the k-piece
packing problem. Due to the k-surplus of A
k
, each vertex of A
k
has degree at least k +1
in G
k
. Because the new vertices of G
k
(ie. the vertices in V (G
k
) − V (G)) have degree at
most u(v) − l(v)+1≤ k,wegetthatA
k
⊆ V (G). So the deletion of the new vertices
yields a partition V (G)=D
˙
∪ A
˙

∪ C where D = D
k
∩V (G),A= A
k
and C = C
k
∩V (G).
This canonical partition has all the properties listed in Theorem 2.8, for example the
connected components of G[D]are(l, u)-galaxies, for all ∅= A

⊆ A the number of those
(l, u)-galaxy components of G[D] which are adjacent to A

is at least u(A

) + 1, and C has
a perfect (l, u)-piece packing. This partition is unique, because if V (G)=D

˙
∪ A

˙
∪ C

is another partition with these properties then in G
k
the set A

is a perfect barrier with
k-surplus, hence by Corollary 4.11 it equals to A

k
. The analogue of Theorem 4.19 also
holds.
This Edmonds-Gallai type theorem for the (l, u)-piece packing problem becomes quite
compact in the case l(v)=l<u= u(v) for all v ∈ V (G), so we include this. Here an
the electronic journal of combinatorics 12 (2005), #R8 20
(l, u)-piece packing is a packing with connected graphs F with l ≤ ∆(F) ≤ u.Callsuch
a packing an (l<u)-packing. The simplicity of this structure theorem comparing to the
general case is due to the fact that here an (l, u)-galaxy is just a graph with highest degree
at most l − 1. So it always consists of only one tip.
Theorem 8.2. For a graph G let D = {v ∈ V (G): v can be missed by a maximal (l<u)-
packing of G}.LetA =Γ(D) and C = V (G) − (D ∪ A).Now
1. ∆(G[D]) ≤ l − 1,
2. for all ∅= A

⊆ A the number of those components of G[D] which are adjacent to
A

is at least u|A

| +1,
3. G[C] has a perfect (l<u)-packing, and
4. for each maximal (l<u)-packing P of G, the graph G−P has exactly c(G[D])−u|A|
connected components.
References
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